Well, sort of. If you use only part of the sensor — as you do in DX crop mode — you are inherently gathering less light. (The light per area is the same — so your exposure settings are the same — but you have less area.) That means that if you print at the same size as you would an image from an FX lens, your print inherently has less light per area — or, to think of it another way, any noise would be enlarged more than it would be otherwise, and theoretically more apparent.

Of course, this is also true if you crop some area from an image from your FX lens and enlarge. In most practical terms, the difference is so small that no one ever thinks about it.

The noise will only be more apparent if the exposure is lower, no matter how much light is in the environment. If the aperture and shutter time is adjusted appropriately the noise characteristics are theoretically the same, ignoring sources of noise from long exposure etc.
– damned truthsMar 24 '15 at 5:49